Ha, interesting collection of ideas. I like the "flying under the radar" goal, especially with the story of the player being a psychic trying to avoid detection.

Perhaps in order to provide a progression curve, or difficulty feature, the player’s level of perception would increase or decrease depending on factors like stress. I.e. the number of other cards the player can see depends upon the amount of risk, the stress the player is under depending on how much the other players suspect foul play. That way, the riskier the play is the harder it is for them to reliably know the outcome, or that if they play with less cards visible, the risk is reduced and they can play it safer. Like the player has a choice of how many cards they want to reveal using their powers, and this directly influences the risk of being caught. Something like that.